Average Error: 0.2 → 0.1
Time: 8.3s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot \left(\frac{\sqrt{x} - \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot 6\right)\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot \left(\frac{\sqrt{x} - \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot 6\right)
double f(double x) {
        double r1072372 = 6.0;
        double r1072373 = x;
        double r1072374 = 1.0;
        double r1072375 = r1072373 - r1072374;
        double r1072376 = r1072372 * r1072375;
        double r1072377 = r1072373 + r1072374;
        double r1072378 = 4.0;
        double r1072379 = sqrt(r1072373);
        double r1072380 = r1072378 * r1072379;
        double r1072381 = r1072377 + r1072380;
        double r1072382 = r1072376 / r1072381;
        return r1072382;
}
↓
double f(double x) {
        double r1072383 = x;
        double r1072384 = sqrt(r1072383);
        double r1072385 = 1.0;
        double r1072386 = sqrt(r1072385);
        double r1072387 = r1072384 + r1072386;
        double r1072388 = 4.0;
        double r1072389 = r1072383 + r1072385;
        double r1072390 = fma(r1072384, r1072388, r1072389);
        double r1072391 = sqrt(r1072390);
        double r1072392 = r1072387 / r1072391;
        double r1072393 = r1072384 - r1072386;
        double r1072394 = r1072393 / r1072391;
        double r1072395 = 6.0;
        double r1072396 = r1072394 * r1072395;
        double r1072397 = r1072392 * r1072396;
        return r1072397;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.1
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{6}}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.0

    \[\leadsto \frac{x - 1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{\color{blue}{1 \cdot 6}}}\]

  5. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{x - 1}{\frac{\color{blue}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)} \cdot \sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}}{1 \cdot 6}}\]

  6. Applied times-frac0.3

    \[\leadsto \frac{x - 1}{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}}\]

  7. Applied add-sqr-sqrt0.3

    \[\leadsto \frac{x - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]

  8. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{1} \cdot \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]

  9. Applied difference-of-squares0.2

    \[\leadsto \frac{\color{blue}{\left(\sqrt{x} + \sqrt{1}\right) \cdot \left(\sqrt{x} - \sqrt{1}\right)}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]

  10. Applied times-frac0.1

    \[\leadsto \color{blue}{\frac{\sqrt{x} + \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{1}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}}\]

  11. Simplified0.1

    \[\leadsto \color{blue}{\frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}} \cdot \frac{\sqrt{x} - \sqrt{1}}{\frac{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}{6}}\]
  12. Using strategy rm

  13. Applied associate-/r/0.1

    \[\leadsto \frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot \color{blue}{\left(\frac{\sqrt{x} - \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot 6\right)}\]

  14. Final simplification0.1

    \[\leadsto \frac{\sqrt{x} + \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot \left(\frac{\sqrt{x} - \sqrt{1}}{\sqrt{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}} \cdot 6\right)\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"
  :precision binary64

  :herbie-target
  (/ 6 (/ (+ (+ x 1) (* 4 (sqrt x))) (- x 1)))

  (/ (* 6 (- x 1)) (+ (+ x 1) (* 4 (sqrt x)))))